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# Class Note for ECE 6340 with Professor Jackson at UH

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Date Created: 02/06/15

ECE 6340 Intermediate EM Waves Fall 2008 Prof David R Jackson Dept of ECE Notes 16 Polarization of Waves Consider a plane wave with both x and y components Assume Ex a real number Ey bem In general 8 phase of Ey phase of Ex Polarization of Waves cont Time Domain 0 At Z 0 lt5 lt5 Re elem a cos wt Re 9613 em b cosat Depending on b and three different cases arise Linear Polarization Ex At 2 O 5 Reaejmacosat c5 Reib em 2 ii cos wt shown for 0 Circular Polarization b a AND zi 7r2 At Z 0 5 Reaej acoscot 5 Rebej ejm ReaeiJ Zejm acosat i 7r2 61 sin wt 2 l l 52 fv2 612 cos2 cot 612 sm2 cat 2 61 Circular Polarization cont Circular Polarization cont IEEE convention y 0 8 7z2 a LHCP 8 7rik RHCP Circular Polarization cont RHCP RHCP 39 cot k2 e opposite rotation in space and time Unit Rotation Vectors 2 i6c jb J5 A 1 A A 2 x JEL 13 A 1 A A Add x D J5 S 39 Aquot j A A ubtract Z J5 RHCP LHCP 10 General Wave Representation EngxiEy 1 1 Ex lt ZgtEy 2 2 J5 J5 Note any polarization can be written as combination of RHCP and LHCP waves 11 Elliptical Polarization Includes all other cases 7 0 or it and i 7r2 or W i 7z2 and b a y quot 0 1 I I I I x I I I I I l I 12 Elliptical Polarization cont RHEP 13 Proof of Ellipse Property 5 bcosat 5 acosat bcoswtcos bsinwtsin SO ibcos bsin 1 j a a b b 2 i jccos s1n bZ j 62 a iz2 cos Z Ecos jzsin23b2 62 61 a 14 Proof of Ellipse Property cont 9 2 b b 2 iz jCZ cos Z cos sin2 b2 j 62 a a a 6 cos2 sin2 62 2amp6 cos B 92 sin2 B a b 2 b 52 ZZoos 8 5va 92 sin2 8 General quadratic form A jfB jc oyC iz D Proof of Ellipse Property cont A B2 4AC cos2 a a ET cos2 1 a b 2 so A 4 j sin2 lt0 61 Hence this is an ellipse 16 Rotation Property lt5 lt5 61 cos wt 9 cosat 8 boosmtcos bsinmtsin 0 x 17 Rotation Property cont 50 tang5 zgyz hos tanwtsin x 61 Take the derivative sec2 sec2 wtasin 3 so sin BKEJCOSZ secz mm a a Olt lt7z gw LHEP 9 7r lt lt 0 g gt 0 RHEP proof complete 18 Phasor Picture Rule The electric field vector rotates from the leading axis to the lagging axis 19 Phasor Picture cont Rule The electric field vector rotates from the leading axis to the lagging axis Example lt1jgtilt2 jgtejky What is this wave s polarization 21 Example cont E 1ji2jejky EZ leads Ex Re In time the wave rotates from the Z axis to the x axis 22 Note Example cont E 1ji2jejky LHEP or LHCP E x E Z 7 7t and i ii so this is not LHCP Example cont Eilt1jamp2 jgte y LH EP 24 Axial Ratio AR and Tilt Angle r major axis AB AR mlnoraXIs CD gt1 25 Axial Ratio AR and Tilt Angle r 7 tan 1 2 Axial Ratio a 0373900 sm2fsm2ysm 45 g g g 45 Tilt Angle AR 2 i cot 95 tan 2239 tan 27 cos 8 for LHEP for RHEP Note The tilt angle 1 is ambiguous to the addition of i 90 26 Axial Ratio AR and Tilt Angle r Tilt Angle tan 2239 tan 27 cos 8 chzacosmt bcosati7r2 19 sin at I

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